Row Reduction Algorithm
Pivot
A pivot position in a matrix is a location in that corresponds to a leading entry (a leading "1") when is row reduced to a matrix in echelon form (reduced echelon form).
A pivot column is a column that contains a pivot position.
A pivot is a nonzero number in a pivot position that is used as needed to create zeros in the column that contains that nonzero number through the use of replacement row operations.
We will introduce a systematic method called the row reduction algorithm to locate all the pivot positions one by one and transform the matrix to the one in echelon form (and reduced echelon form).
Row Reduction Algorithm
The row reduction algorithm consists of four steps, and it produces a matrix in echelon form. The last step produces a matrix in reduced echelon form.
Step 1: Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top.
Step 2: Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position.
Step 3: Use row replacement operations to create zeros in all positions below the pivot.
Step 4: Ignore the row containing the pivot position and all rows, if any, above it. Apply steps 1-3 to the submatrix that remains. Repeat the process until there are no more nonzero rows to modify.
Example: Given the following augmented matrix
We will use the "row operation calculator" again to demonstrate the row reduction algorithm.
Exercise:
Use the row reduction algorithm to transform the following augmented matrix into the one in echelon form: